Subgroups — C₂²×C₃₇×#384 #22655263

Order 9472.

5502 conjugacy classes of subgroups.

The subgroup lattice has 126 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊C₄×C₃₇	C₂	1	6	yes	maximal
2	C₂³×C₄²×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₇×#384	C₂	1	24	yes	maximal, direct product factor
4	C₂⁴×C₄×C₃₇	C₂²	1	4	yes	—
5	C₂⁴×C₄×C₃₇	C₂²	1	3	yes	—
6	C₂²×C₄⋊C₄×C₃₇	—	2	48	no	—
7	C₂²×C₄⋊C₄×C₃₇	C₂²	1	72	yes	—
8	C₂²×C₄²×C₃₇	C₄	1	16	yes	complemented
9	C₂²×C₄²×C₃₇	C₂²	1	12	yes	—
10	C₃₇×#384	C₂²	1	64	yes	direct product factor
11	C₂⁵×C₃₇	C₂³	1	1	yes	characteristic
12	C₂³×C₄×C₃₇	—	2	48	no	—
13	C₂³×C₄×C₃₇	C₂×C₄	1	24	yes	—
14	C₂³×C₄×C₃₇	D₄	1	24	yes	—
15	C₂³×C₄×C₃₇	Q₈	1	24	yes	—
16	C₂³×C₄×C₃₇	C₂³	1	24	yes	—
17	C₂³×C₄×C₃₇	C₂³	1	18	yes	—
18	C₂×C₄⋊C₄×C₃₇	—	2	288	no	—
19	C₂×C₄⋊C₄×C₃₇	C₂³	1	96	yes	—
20	C₂×C₄²×C₃₇	C₂×C₄	1	96	yes	complemented
21	C₂×C₄²×C₃₇	C₂³	1	16	yes	—
22	C₂⁴×C₃₇	C₂²×C₄	1	4	yes	—
23	C₂⁴×C₃₇	C₂×D₄	1	12	yes	—
24	C₂⁴×C₃₇	C₂×Q₈	1	12	yes	—
25	C₂⁴×C₃₇	C₂⁴	1	3	yes	—
26	C₂²×C₄×C₃₇	—	2	144	no	—
27	C₂²×C₄×C₃₇	—	4	64	no	—
28	C₂²×C₄×C₃₇	C₄⋊C₄	1	192	yes	complemented
29	C₂²×C₄×C₃₇	C₂²×C₄	1	72	yes	—
30	C₂²×C₄×C₃₇	C₂×D₄	1	72	yes	—
31	C₂²×C₄×C₃₇	C₂×Q₈	1	72	yes	—
32	C₂²×C₄×C₃₇	C₂⁴	1	16	yes	—
33	C₂²×C₄×C₃₇	C₂⁴	1	12	yes	—
34	C₄⋊C₄×C₃₇	—	2	192	no	—
35	C₄²×C₃₇	C₂²×C₄	1	64	yes	complemented
36	C₂³×C₃₇	C₂×C₄⋊C₄	1	48	yes	—
37	C₂³×C₃₇	#127	1	16	yes	—
38	C₂³×C₃₇	C₄⋊Q₈	1	48	yes	—
39	C₂³×C₃₇	C₂³×C₄	1	6	yes	—
40	C₂³×C₃₇	C₂²×D₄	1	18	yes	—
41	C₂³×C₃₇	C₂²×Q₈	1	18	yes	—
42	C₂³×C₃₇	C₂⁵	1	1	yes	characteristic
43	C₂×C₄×C₃₇	—	2	48	no	—
44	C₂×C₄×C₃₇	—	4	96	no	—
45	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	288	yes	complemented
46	C₂×C₄×C₃₇	C₂³×C₄	1	24	yes	—
47	C₂×C₄×C₃₇	C₂²×D₄	1	24	yes	—
48	C₂×C₄×C₃₇	C₂²×Q₈	1	24	yes	—
49	C₂²×#384	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂²×C₃₇	#384	1	64	yes	direct product factor
51	C₂²×C₃₇	C₂²×C₄⋊C₄	1	36	yes	—
52	C₂²×C₃₇	C₂×#127	1	12	yes	—
53	C₂²×C₃₇	C₂×C₄⋊Q₈	1	36	yes	—
54	C₂²×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
55	C₂²×C₃₇	C₂³×D₄	1	3	yes	—
56	C₂²×C₃₇	C₂³×Q₈	1	3	yes	—
57	C₄×C₃₇	—	4	16	no	—
58	C₄×C₃₇	C₂²×C₄⋊C₄	1	48	yes	complemented
59	C₂³×C₄⋊C₄	C₂×C₃₇	1	6	yes	—
60	C₂³×C₄²	C₂×C₃₇	1	1	yes	characteristic
61	C₂×#384	C₂×C₃₇	1	24	yes	direct product factor
62	C₂×C₃₇	C₂×#384	1	24	yes	direct product factor
63	C₂×C₃₇	C₂³×C₄⋊C₄	1	3	yes	—
64	C₂×C₃₇	C₂²×#127	1	1	yes	characteristic
65	C₂×C₃₇	C₂²×C₄⋊Q₈	1	3	yes	—
66	C₂⁴×C₄	C₂²×C₃₇	1	4	yes	—
67	C₂⁴×C₄	C₂²×C₃₇	1	3	yes	—
68	C₂²×C₄⋊C₄	—	2	48	no	—
69	C₂²×C₄⋊C₄	C₂²×C₃₇	1	72	yes	—
70	C₂²×C₄²	C₄×C₃₇	1	16	yes	complemented
71	C₂²×C₄²	C₂²×C₃₇	1	12	yes	—
72	#384	C₂²×C₃₇	1	64	yes	direct product factor
73	C₃₇	C₂²×#384	1	1	yes	characteristic, direct product factor, Sylow
74	C₂⁵	C₂³×C₃₇	1	1	yes	characteristic
75	C₂³×C₄	—	2	48	no	—
76	C₂³×C₄	C₂×C₄×C₃₇	1	24	yes	—
77	C₂³×C₄	D₄×C₃₇	1	24	yes	—
78	C₂³×C₄	Q₈×C₃₇	1	24	yes	—
79	C₂³×C₄	C₂³×C₃₇	1	24	yes	—
80	C₂³×C₄	C₂³×C₃₇	1	18	yes	—
81	C₂×C₄⋊C₄	—	2	288	no	—
82	C₂×C₄⋊C₄	C₂³×C₃₇	1	96	yes	—
83	C₂×C₄²	C₂×C₄×C₃₇	1	96	yes	complemented
84	C₂×C₄²	C₂³×C₃₇	1	16	yes	—
85	C₂⁴	C₂²×C₄×C₃₇	1	4	yes	—
86	C₂⁴	C₂×D₄×C₃₇	1	12	yes	—
87	C₂⁴	C₂×Q₈×C₃₇	1	12	yes	—
88	C₂⁴	C₂⁴×C₃₇	1	3	yes	—
89	C₂²×C₄	—	2	144	no	—
90	C₂²×C₄	—	4	64	no	—
91	C₂²×C₄	C₄⋊C₄×C₃₇	1	192	yes	complemented
92	C₂²×C₄	C₂²×C₄×C₃₇	1	72	yes	—
93	C₂²×C₄	C₂×D₄×C₃₇	1	72	yes	—
94	C₂²×C₄	C₂×Q₈×C₃₇	1	72	yes	—
95	C₂²×C₄	C₂⁴×C₃₇	1	16	yes	—
96	C₂²×C₄	C₂⁴×C₃₇	1	12	yes	—
97	C₄⋊C₄	—	2	192	no	—
98	C₄²	C₂²×C₄×C₃₇	1	64	yes	complemented
99	C₂³	C₂×C₄⋊C₄×C₃₇	1	48	yes	—
100	C₂³	#127×C₃₇	1	16	yes	—
101	C₂³	C₄⋊Q₈×C₃₇	1	48	yes	—
102	C₂³	C₂³×C₄×C₃₇	1	6	yes	—
103	C₂³	C₂²×D₄×C₃₇	1	18	yes	—
104	C₂³	C₂²×Q₈×C₃₇	1	18	yes	—
105	C₂³	C₂⁵×C₃₇	1	1	yes	characteristic
106	C₂×C₄	—	2	48	no	—
107	C₂×C₄	—	4	96	no	—
108	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	288	yes	complemented
109	C₂×C₄	C₂³×C₄×C₃₇	1	24	yes	—
110	C₂×C₄	C₂²×D₄×C₃₇	1	24	yes	—
111	C₂×C₄	C₂²×Q₈×C₃₇	1	24	yes	—
112	C₂²	C₃₇×#384	1	64	yes	direct product factor
113	C₂²	C₂²×C₄⋊C₄×C₃₇	1	36	yes	—
114	C₂²	C₂×#127×C₃₇	1	12	yes	—
115	C₂²	C₂×C₄⋊Q₈×C₃₇	1	36	yes	—
116	C₂²	C₂⁴×C₄×C₃₇	1	1	yes	characteristic
117	C₂²	C₂³×D₄×C₃₇	1	3	yes	—
118	C₂²	C₂³×Q₈×C₃₇	1	3	yes	—
119	C₄	—	4	16	no	—
120	C₄	C₂²×C₄⋊C₄×C₃₇	1	48	yes	complemented
121	C₂	C₂×C₃₇×#384	1	24	yes	direct product factor
122	C₂	C₂³×C₄⋊C₄×C₃₇	1	3	yes	—
123	C₂	C₂²×#127×C₃₇	1	1	yes	characteristic
124	C₂	C₂²×C₄⋊Q₈×C₃₇	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₃₇	C₂³
commutator	C₂²	C₂⁴×C₄×C₃₇
Frattini, cosocle	C₂³	C₂⁵×C₃₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₃₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂×C₃₇×#384	C₂	direct (N × H)
8. C₂²×C₄²×C₃₇	C₄	semidirect
10. C₃₇×#384	C₂²	direct (N × H)
20. C₂×C₄²×C₃₇	C₂×C₄	semidirect
28. C₂²×C₄×C₃₇	C₄⋊C₄	semidirect
35. C₄²×C₃₇	C₂²×C₄	semidirect
45. C₂×C₄×C₃₇	C₂×C₄⋊C₄	semidirect
49. C₂²×#384	C₃₇	direct (N × H)
50. C₂²×C₃₇	#384	direct (N × H)
58. C₄×C₃₇	C₂²×C₄⋊C₄	semidirect
61. C₂×#384	C₂×C₃₇	direct (N × H)
62. C₂×C₃₇	C₂×#384	direct (N × H)
70. C₂²×C₄²	C₄×C₃₇	semidirect
72. #384	C₂²×C₃₇	direct (N × H)
73. C₃₇	C₂²×#384	direct (N × H)
83. C₂×C₄²	C₂×C₄×C₃₇	semidirect
91. C₂²×C₄	C₄⋊C₄×C₃₇	semidirect
98. C₄²	C₂²×C₄×C₃₇	semidirect
108. C₂×C₄	C₂×C₄⋊C₄×C₃₇	semidirect
112. C₂²	C₃₇×#384	direct (N × H)
120. C₄	C₂²×C₄⋊C₄×C₃₇	semidirect
121. C₂	C₂×C₃₇×#384	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₄⋊C₄×C₃₇	2
C₂³×C₄²×C₃₇	2
C₂×C₃₇×#384	2
C₂²×#384	37

Subgroups — C22×C37×#384 #22655263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₇×#384 #22655263